Why does the hydrolysis of 2-iodobutane give a single optical isomer as organic product whereas the hydrolysis of 2-methyl-2-iodohexane gives a mixture of 2 optical isomers?

I don't see it. Are you sure the names and info is correct?

I'm assuming by hydrolysis you are talking about substitution of the halide with OH.

2-iodobutane would produce an optically active isomer in forming 2-butanol. But id expect a polar, protic nucleophile to force SN1 products... granted, it's a secondary alkyl halide, but you'd still get at least a mixture of both R/S isomers. Its common to see both sn1 and sn2 products with secondary r-x's. So unless I'm missing something, and maybe I am, I think you'd see two optically active isomers of 2-butanol for the first case.

the second one also confuses me a bit. Assuming again that you do a substitution reaction on this compound, a tertiary alkyl halide, you'd definitely expect a carbocation mechanism in a protic solvent... but... the reaction site isn't chiral. There are two methyl groups off the second carbon... so after the h2o adds, there is no chiral product at all. Again, maybe I'm missing something.

do you have more info about this question or the answer?